Branching of hydraulic cracks enabling permeability ofgas or oil shale with closed natural fracturesSaeed Rahimi-Aghdama, Viet-Tuan Chaub, Hyunjin Leea, Hoang Nguyena, Weixin Lia, Satish Karrab, Esteban Rougierb,Hari Viswanathanb, Gowri Srinivasanc, and Zdenek P. Bazanta,d,e,1

aDepartment of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208; bEarth and Environmental Sciences Division, Los AlamosNational Laboratory, Los Alamos, NM 87545; cX Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545; dDepartment ofMechanical Engineering, Northwestern University, Evanston, IL 60208; and eDepartment of Materials Science, Northwestern University, Evanston, IL 60208

Contributed by Zdenek P. Bazant, November 21, 2018 (sent for review October 29, 2018; reviewed by Huajian Gao and John Hutchinson)

While hydraulic fracturing technology, aka fracking (or fraccing,frac), has become highly developed and astonishingly successful,a consistent formulation of the associated fracture mechanics thatwould not conflict with some observations is still unavailable. Itis attempted here. Classical fracture mechanics, as well as currentcommercial software, predict vertical cracks to propagate with-out branching from the perforations of the horizontal well casing,which are typically spaced at 10 m or more. However, to explainthe gas production rate at the wellhead, the crack spacing wouldhave to be only about 0.1 m, which would increase the overallgas permeability of shale mass about 10,000×. This permeabilityincrease has generally been attributed to a preexisting system oforthogonal natural cracks, whose spacing is about 0.1 m. How-ever, their average age is about 100 million years, and a recentanalysis indicated that these cracks must have been completelyclosed by secondary creep of shale in less than a million years.Here it is considered that the tectonic events that produced thenatural cracks in shale must have also created weak layers withnanocracking or microcracking damage. It is numerically demon-strated that seepage forces and a greatly enhanced permeabilityalong the weak layers, with a greatly increased transverse Biotcoefficient, must cause the fracking to engender lateral branchingand the opening of hydraulic cracks along the weak layers, even ifthese cracks are initially almost closed. A finite element crack bandmodel, based on a recently developed anisotropic spherocylin-drical microplane constitutive law, demonstrates these findings[Rahimi-Aghdam S, et al. (2018) arXiv:1212.11023].

fracking | poromechanics | Biot coefficient | seepage forces | damage

S ignificant advances have been made in fracture mechanicsof propagation of a single hydraulic crack in elastic rock

under tectonic stress (1–11). They include characterization ofthe stress singularity at the tip of a water-filled advancing crack;flow of water of controlled viscosity along the crack, with orwithout proppant grains; and water leak-off into the shale.Interactions of parallel cracks, their stability, closing, and stressshadow effect have also been clarified (12–15). Discrete elementmodels, in which the hydraulic crack was simulated by a band ofinterelement separations (16, 17), led to similar results.

These studies, however, predicted no branching of the hy-draulic cracks, originally spaced at cca 10 m. This presented adilemma since branching is the only way to reduce the crackspacing to about 0.1 m, which is necessary to explain the gasproduction rate. Consequently, it has been universally hypoth-esized that the preexisting natural cracks, spaced at ca. 0.1 m,would somehow increase the overall permeability of the shalemass. A 10,000-fold increase of permeability would be neces-sary to match the gas production rate. However, recent analysis(18, 19) showed that the natural, tectonically produced, cracks,which are, on the average, about 108 years old, must have beenclosed by secondary creep (or viscous flow) of shale under tec-tonic stress within 104 years to 106 years (if not filled earlier bycalcite deposit). This invalidated the hypothesis.

It might be objected that water in the cracks could have pre-vented crack closing. However, the open spaces in shear cracks,created (due to shear dilatancy) by a tectonic event, could not

have been filled by water immediately. If the water had to seepin from the ground surface, it would take about 10 million yearsand, if from a nearby water-filled rock formation, certainly overa million years. This must have left plenty of time for the creepclosing to proceed uninhibited.

A recent paper (20) presented a new model which, by con-trast with all of the previous studies, took into account (i) theseepage forces (i.e., the body forces due gradients of pore pres-sure in Darcy diffusion of water into porous shale) and (ii) thevariation of effective Biot coefficient for the water pressure onthe crack plane, caused by gradually vanishing bridges betweenthe opposite faces of a widening bridged crack (another differ-ence from the previous studies was abandoning the assumptionof incompressibility of water in the cracks, since water is about 20times more compressible than shale). This model (20) did predictextensive lateral crack branching.

Later analysis, however, showed that the branching indicatedby the computer program in ref. 20 was, in fact, triggered bythe unintended coding of a sudden change of Biot coefficientfor transverse water pressure on the crack. This change abruptlyincreased the water pressure on the solid phase and triggereddynamic response. Such a sudden trigger is probably unrealistic,which represents a vital correction to the preceding study (20)(this correction nevertheless reveals a useful fact, namely, thatfluid pressure shocks could greatly enhance crack branching).

If Biot coefficient is changed gradually, the model from theprevious study (20) would predict no crack branching, althoughthe branching must occur to explain the observed gas produc-tion rate. This study will show that, if the previous model (20)is enhanced by introducing, into the shale mass, significant het-erogeneity due to damaged weak layers along preexisting naturalcracks, then an extensive and dense crack branching is predicted.

It may be noted that the fracking companies are aware ofthe necessity of branched cracks running along preexisting nat-ural fractures. Fig. 1 shows a picture similar to what is foundon the websites of some companies. However, this awarenessseems to be merely intuitive and empirical. The existing com-mercial software, as well as fracture mechanics studies, predicts

Significance

Development of a realistic model of fracking would allow bet-ter control. It should make it possible to optimize variousparameters such as the history of pumping, its rate or cycles,changes of viscosity, etc. This could lead to an increase of thepercentage of gas extraction from the deep shale strata, whichcurrently stands at about 5% and rarely exceeds 15%.

Author contributions: S.R.-A. and Z.P.B. designed research and conceived the mathemat-ical model; S.R.-A., V.-T.C., H.L., H.N., W.L., S.K., E.R., H.V., G.S., and Z.P.B. performedresearch; S.R.-A. analyzed data; and S.R.-A. and Z.P.B. directed the research and wrotethe paper.yReviewers: H.G., Brown University; and J.H., Harvard University. y

The authors declare no conflict of interest.y

Published under the PNAS license.y1 To whom correspondence should be addressed. Email: z-bazant@northwestern.edu.y

Published online January 11, 2019.

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Fig. 1. Schematic branching due to natural fractures. (A) Water is injectedat high pressure through damaged zones and weak layers, (B) crack branch-ing initiates due to the presence of damaged zones and natural fractures,and (C) dense cracking happens in all directions, due to the presence of dam-aged zones, weak layers at closed natural fractures (downward view normalto bedding plane).

no branching. So an intersecting system of open natural frac-tures is assumed to either exist a priori or to develop according tosome empirical criteria with no basis in mechanics, supported bysome recent experiments indicating the possibility of branching(21–24). A physics-based model for branching, which is our goal,seems lacking.

Fluid Flow in Porous Solid, Without or With Cracking DamageTwo types of flow play a role in hydraulic fracturing: (i) theflow along the hydraulically created cracks, typically a few mil-limeters wide, and (ii) the flow through nanoscale pores andmicrocracks or nanocracks in shale with preexisting damage. Thelatter is negligible after continuous hydraulic cracks form, buthere it is found to be crucially important for crack initiation andbranching. The volume flow, q, of water through the pores andnanocracks or microcracks of isotropic material may be approxi-mately calculated from the Darcy law: q=−(K/µ)∇ψ, where Kis the permeability, µ is the dynamic viscosity, and ψ is the phasepotential calculated as ψ= p− γgz . Here p is the pore pressure,γg is the pressure gradient due to gravity, and z is the depthfrom a datum. However, the permeability Kv in the directionnormal to the bedding planes (x , y), i.e., in vertical direction z ,

is much lower than permeability Kh along these planes (horizon-tal). Therefore, the 3D Darcy law is, in general, anisotropic. InCartesian coordinates (x , y , z ), the resulting volume flux vectorq=(qx , qy , qz ) may be written as

q=−µ−1K ·∇ψ, [1]

where ∇ is the vector of gradient operator; K is the 3× 3 per-meability matrix, which is diagonal if (and only if) the Cartesianaxes x , y , z are chosen to be parallel and normal to the beddingplanes.

Although the natural (or preexisting) cracks in shale strata at3 km depth must have been closed by 100 million years of creep,the damage bands along these cracks, which always accompanypropagation of fracture process zone, certainly remain (in fact,based on the known surface energy of shale, it can be shownthat even empty pores and cracks ca. <15 nm in size, at depth3 km, cannot close, and this is confirmed by the known size ofpores containing shale gas). Permeability Kxx along these bandsis surely much higher than it is in the intact shale (although poresof <15 nm contribute nothing globally).

To prevent the formation of horizontal cracks, the pump-ing pressure is assumed not to exceed the overburden pressure,which is about 75 MPa. Hydraulic fracturing is considered to pro-duce a system of mutually orthogonal vertical cracks, normal tothe directions of the minimum and maximum principal tectonicstresses. The flow of the second type, along the hydraulically cre-ated cracks, may be assumed to follow the Reynolds equationsof classical lubrication theory, which are based on the Poiseuillelaw for viscous flow. Thus, the horizontal and vertical flow vec-tor components in x , y , z directions along with the cracks may becalculated as

Qx =−h2y

12 µ∇xp, Qy =−

h2x

12 µ∇yp, Qz =−

h2x + h2

y

12 µ∇zp,

[2]

A

B

Fig. 2. (A) Fluid flow in intact and damaged shale. (B) Biot coefficientincrease in transverse direction due to increasing damage (20).

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where ∇x = ∂/∂x ,. . .; hx , hy are the opening widths of verti-cal cracks normal to axes x and y that are positioned into thebedding plane.

An effective way to simulate the hydraulic cracks numericallyis the crack band model (25–27), in which cracking deformationis considered smeared over the band (or element) width. Thewidths of cracks normal to x and y are (Fig. 2)

hx = lx ε”xx , hy = lyε”yy , [3]

where ε”xx , ε”yy are damage parts of normal strains due tosmeared cracking normal to x and y directions; lx , ly are thewidths of crack bands, assumed equal to the minimum pos-sible spacing of adjacent parallel hydraulic cracks (lx , ly mustbe treated as a material property, related to fracture energyGf of shale; here lx , ly are not changed but, if they were, thepostpeak softening would have to be adjusted to preserve Gf ).Furthermore,

ε”ij = εij − εelij , εelij =Cijklσkl , [4]

where Cijkl is the transversely isotropic elastic compliance tensorof shale (for unloading); σij , εij are the stress and strain tensorsin the rock, calculated from a constitutive model for smearedcracking damage [with a localization limiter (25)], for which thespherocylindrical microplane constitutive model (28) has beenused. The coordinates are Cartesian, xi , i =1, 2, 3 (x1≡ x , x2≡y , x3≡ z ). Note that, the same as in ref. 20, water is consideredas compressible. It is, in fact, about 20 times more compressiblethan concrete, and the water pressure during fracking can be high(up to 70 MPa).

Equilibrium in Two-Phase Solid and Biot CoefficientThe shale may be modeled as a two-phase medium with water-saturated pores, for which the classical Biot-type relations for theequilibrium of the phases apply. For undamaged shale, they read

Sij =σij − δij b0p, [5]

where p is the pore pressure, b0 is the Biot coefficient of undam-aged shale, Sij is the total stress tensor, σij is the stress tensor inthe solid phase, and δij is the Kronecker delta. As a special case,SV =σV − b0p, where SV =Skk/3 is the volumetric total stress,and σij =σkk/3 is the volumetric stress in the solid phase.

0

0.2

0.4

0.6

0.8

1

0 1 2 3

Biot

Coe

ffici

ent

Time0

0.4

0.8

1.2

1.6

2

0 1 2 3

Stre

ss (M

Pa)

Time

Initial damage

Strain Pressure

x

y Injection point

D

A B C

E

Fig. 3. Results of two-phase FE simulations in case of a single damage band.(A) Initial model. (B) Strain εxx due to water injection (the red marks thehighest values; the blue marks the smallest values). (C) Pressure propaga-tion. (D) Stress evolution vs. injection time for the first element above theinitially damaged elements. (E) Evolution, along the damage band, of theBiot coefficient transverse to the band.

Ini�al crack

-60

-50

-40

-30

-20

-10

0

0 2 4 6 8

Stre

ss, σ

x(M

Pa)

Time

No seepage forceStrain

x

y

Stress of this element is analyzed

A

C

B

D

Fig. 4. (A) Pressurized line crack in a 2D domain of a two-phase porous solid(shale) supported by springs at boundaries, subjected to tectonic stresses Tx

and Ty . (B) FE mesh for one-half of the domain. (C) Extension of the bandof high strain. (D) Evolution of stress σxx in solid part in the element at thecenter of initial crack face.

While, typically, ϕ=0.1, the Biot coefficient of shales can varybetween 0.2 and 0.7. Test results (29–34) show that it increaseswith the cracking damage and depends on the load direction.This requires generalizing the Biot coefficient as a tensor, bij(35). The following, tensorially consistent, empirical relation,which appears to match test data, is proposed:

bij =min{b0 +βε”ij (ε”kk/3)

−2/3, 1}

(ϕ≤ b0≤ 1). [6]

Here b0 refers to undamaged material, β is an empirical param-eter, ε′′ij is the inelastic damage strain tensor, and ϕ is thenatural porosity of shale. For the Biot coefficient in the direc-tion of unit vector νi , this equation gives bν = νiνj bij = b0 +

βε”ν(ε”V )−2/3 (but≤1), where ε”V = ε”kk/3 is the inelastic rel-ative volume expansion, and ε”ν = νiνj ε”ij is the inelastic normalstrain component in direction of vector νi .

For the special case of microcracking or nanocracking nor-mal to x1 direction only, one has ε”V = ε”11/3 and bν = b0 +

β(9ε”11)1/3 (but ≤1). This equation can be interpreted graph-

ically as seen in Fig. 2B, which shows section A–A of a bandof preexisting, mostly aligned, microcracks and the compressivestresses applied by the pore fluid onto the microcrack faces,resisted by tensile stresses in the ligaments of the solid betweenthe microcrack tips.

The viscous drag of water flowing through a soil imposes aseepage force on the soil in the direction of flow. The seepageforces are body forces defined as

f s =−b∇p. [7]

The seepage forces are applied on the porous solid and must bebalanced by stresses in the solid. Seepage in an upward directionreduces the effective stress within the soil. When the water pres-sure at a point in the soil is equal to the total vertical stress at thatpoint, the effective stress is zero, and the soil has no frictionalresistance to deformation (36, 37). The seepage force have longbeen considered in geotechnical engineering to assess the risk ofsand liquefaction in cofferdams (38, 39) or under dams. However(except for ref. 20), they have been ignored in previous studies ofhydraulic fracturing, although they do play a crucial role in crackbranching. A poromechanical finite element (FE) code for a

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Fig. 5. (A) Schematic line crack with weak layer and spring boundaries. (B)Strain and damage evolution. (C) Stress in solid part of one element in weaklayer for a shale with b0 = 0.4 and different weak layer permeabilities. (D)Stress in solid part of one element in weak layer for b0 = 0.2 and differentweak layer permeabilities. (E) Stress in solid part of one element in weaklayer for b0 = 0.4 and different relative weak layer strengths. (F) Schematicof seepage forces. (G–I) Evolution of seepage forces.

two-phase solid automatically takes the seepage forces intoaccount in the form of nodal forces.

Two-Phase FE Simulations for a Single Damage BandTo clarify the role of nanocracking or microcracking, considerfirst a horizontal 2D square block of shale of dimensions 1.1 m×1.1 m, supported at the sides by springs approximately equivalentto an infinite medium, as shown in Fig. 3A. Water is injected atthe center of the south side at the constant rate of 2 m3/s. Theanisotropic spherocylindrical microplane model, with the defaultparameters of shale given in ref. 28, is used as the constitutivemodel; lx = ly = 2.1 m. The initial Biot coefficient is b0 =0.4.The tectonic stresses are Tx =−30 MPa and Ty =−30 MPa.

Consider that there is a single preexisting band of nanocracksor microcracks predominantly aligned with axis y , representedby the two red elements in Fig. 3A (which is what remains after acrack was closed by up to a million years of secondary creep, orviscous flow of rock). These cracks cause the vertical permeabil-ity in these two elements to increase ca. 1,000 times comparedwith undamaged shale, while the Biot coefficient increases up to1 and the initial strength decreases to 10% of intact shale.

Fig. 3 B and C shows how damage and pressure propagateafter water injection. For this case, the crack band with highwater pressure is seen to propagate straight forward, withoutbranching. Now look at stress variation. Fig. 3D shows the stressevolution within in the solid part of the first element above theinitial damaged elements. Obviously, the damage during post-peak softening is captured in a stable manner. Finally, considerhow the Biot coefficient and permeability vary in one crackedelement (the first above the initial damaged elements). Fig. 3Econtrasts the evolution of Biot coefficient in the transverse direc-tion with its constancy in the forward direction, which agrees withexperimental observations.

Do the Seepage Forces Suffice to Induce Crack Branching?It is well known in classical fracture mechanics that pressuriz-ing a crack cannot produce tension along the crack faces, andthus cannot initiate lateral crack branching (branching is possi-ble only at the tip of a crack propagating at nearly the Raleighwave speed). In a preceding study (20), it was surmised, undervarious simplifications, that the seepage forces (Eq. 7) wouldsuffice to produce tension along the crack face and thus initi-ate lateral crack branching. Let us examine this more rigorously.Consider again a horizontal 2D square domain 2.5 m × 2.5 m,containing one line crack (Fig. 4A). By virtue of symmetry, onlya half-domain is simulated (Fig. 4B). The water pressure in theline crack is gradually ramped up to reach the maximum of

A B

C D

E F

Elas�c boundary

Shale

MPa

MPa

Ini�alfracked layer

Natural fractures

Injec�onpoint

Fig. 6. FE simulation of hydraulic crack branching in a small domain ofshale with several orthogonal weak layers along perfectly closed natu-ral fractures. (A) Ring of elastic elements providing elastic support of theboundaries. (B) FE mesh, preexisting natural weak layers, and fracking waterinlet. (C–F) Evolution of pressure in a shale with weak layers.

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FE

C D

G

Natural fractures

Injection pressure = 65 MPa

A B

Injec�on points

H

Fig. 7. The 2D FE simulation of fracking process in a horizontal domainwith a larger system of natural fractures or weak layers (the red zone showsthe propagation of high water pressure). (A) FE mesh, injection points, andboundaries. (B) Orthogonal system of preexisting natural fractures. (C–H)Evolution of pressure in a shale with weak layers and initially closed naturalfractures.

50 MPa. Water diffusion from the pressurized crack into theshale is simulated via Darcy law. First, we neglect the increaseof Biot coefficient due to damage (β=0). Fig. 4C shows that thedamage, as well as the crack, propagates only in the direct exten-sion of the initial line crack, i.e., there is no branching. Fig. 4Dshows the evolution of stress in the solid part, σxx , along the crackface. The results show that the Biot coefficient can have a majoreffect and cannot be ignored.

Lateral crack branching would happen if the stress in the solidphase became positive (tensile) and attained the tensile strengthof shale. The results show that this cannot happen, regardlessof the tectonic stress value (even if vanishing). Nevertheless, theseepage forces are seen to reduce the magnitude of compressivestress along the crack face significantly.

We must thus conclude (as an update of ref. 18) that theseepage forces alone do not suffice to explain and model lateralbranching of hydraulic cracks. So, what additional phenomenacould explain the lateral branching? Not surprisingly, the expla-nation is the natural (preexisting) fractures, even though theymust have been completely closed due to millions of years ofsecondary creep, or flow. We demonstrate it next.

Hydraulic Crack Branching in Two-Phase Porous Solid withClosed Natural FracturesIn Fig. 5A, we now consider the same 2D domain of two-phaseporous solid as before, except that now there are two naturalweak layers (or preexisting damage bands) in both x and y direc-tions. The crack is uniformly pressurized, and water diffuses out.The transverse Biot coefficient within the weak layers that rep-resent the closed natural fractures is bnat =1, because the weaklayer (or natural fracture) may consist of separate original crackfaces in contact (uncemented by limestone deposit), while, in theintact shale, the bij values increase according to Eq. 6 from theinitial value b0 =0.4 (Fig. 5C) or 0.2 (Fig. 5D).

Fig. 5B reveals that the hydraulic crack tends to propagatesimultaneously along the initial crack and along the weak layer.This confirms that branching can occur if transverse weak layersexist. Further, consider the normal stress parallel to the crack inone element of the weak layer. If this stress attains the tensilestrength, a lateral crack branch can initiate, and shale branch-ing can happen. Fig. 5B shows the spreading of high and lowertransverse strains along both weak layers for the case of Biotcoefficient b=0.4, with permeability Kweak along the weak band5 times bigger than K0 for intact shale.

The computed effect of ratio Kweak/K0 on the σxx evolution inthe first element of the weak layer above initial crack is plotted inFig. 5 C and D for the initial Biot coefficients, b0 = 0.4 and 0.2. Aswater diffuses into the shale, the stress in the weak layer increasesfrom negative to tensile values until it finally reaches the tensilestrength of the weak layer. Evidently, a greater difference in Biotcoefficient between the weak layer and the shale facilitates, andspeeds up, the crack branching.

Finally, to clarify the effect of the transverse tensile strengthof the weak layer, three relative strength Srel values are con-sidered in Fig. 5E (here Srel is the damaged-to-intact strengthratio of shale). As seen, a smaller Srel leads to smaller stress, but,generally, the effect of Srel is almost negligible. Hence, whetheror not the natural cracks are cemented by limestone is almostirrelevant.

It is instructive to see the evolution of the seepage force vec-tors acting on the mesh nodes, as portrayed in Fig. 5. Fig. 5Fshows schematically the seepage forces acting on an ellipsearound the crack. Fig. 5 G–I illustrates the evolution of seepageforces. Their orientations make it intuitively clear that they mustproduce a biaxial tension in the porous solid at the center of thepressurized domain.

From all these observations, it transpires that a major stimulusfor crack branching is the difference in the Biot coefficient andin the permeability between the weak layers and the intact shale,as well as the shale mass heterogeneity due to the alternation ofweak layers and intact porous solid.

A B C

Fig. 8. (A–C) Water pressure propagation for no weak layers, no naturalfractures.

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It is worth mentioning that the expansion of solid due to theeffect of Biot coefficient has been thought to prevent any tensionparallel to the crack face, and thus cause the closing of any lateralcrack. The preceding results show that this skeptical view doesnot extend to a heterogeneous shale mass containing weak layersalternating with intact shale.

Next consider the horizontal section in Fig. 6A, with 4 weaklayers; bweak =1, Srel =0.1, Kweak/K0 =1000 (Fig. 6B). Wateris injected at one point at constant flow rate. Fig. 6 C–Ethe propagation of high water pressure. Water enters throughprefractured elements, then diffuses along the weak layersand, upon attaining sufficient pressure, the crack branches andPoiseuille flow dominates. The importance of weak layers is thusevidenced.

To demonstrate the present theory on a larger scale, considera bigger horizontal section of shale, a square domain 5 m × 5 m,containing a uniform orthogonal system of closed natural frac-tures with aligned preexisting weak layers (Fig. 7). To be morerealistic, unequal tectonic stresses are considered in x and ydirections; Tx= 30 MPa, and Ty = 40 MPa.

Water is injected at three points at the bottom of Fig. 7A.Fig. 7 C–H shows the evolution of water pressure. The water flowand damage strain are seen to follow the path of weak layers.Extensive branching occurs. Obviously, this branching can cre-ate closely spaced hydraulic cracks and thus increase the overallpermeability of shale stratum by orders of magnitude, comparedwith nonbranching cracks in intact shale.

It has also been checked that omitting the natural fracturesleads to no branching. This is evident from the pressure propaga-tion pattern in Fig. 8. This figure also documents the localizationinstability of parallel crack system (12) (also known as the stressshadow effect), which causes the crack emanating from themiddle injection point not to grow long (the long simultane-ous growth of both remaining cracks is made possible by theproximity of the boundaries).

Conclusionsi) The natural fractures have a major effect on hydraulic frac-

turing and are crucial for its success (although they arecurrently neglected by commercial software).

ii) Even though the natural fractures must have been closedby millions of years of creep, or sealed by mineral deposits,a weak layer of nanocracks and microcracks along thesefractures must be expected to facilitate water diffusion.

iii) Poromechanics with Biot coefficient depending on thedamage of the solid phase must be used in fracking analysis.

iv) Increase of the Biot coefficient in the transverse direction,caused by oriented cracking damage inflicted by fracking, isessential to achieve crack branching.

v) The typical spacing between natural fractures is roughly0.1 m. This matches the hydraulic crack spacing that isnecessary to explain the typical gas production rate at thewellhead.

vi) The widespread opinion that preexisting natural fracturessomehow explain why the overall permeability of shale mass,inferred from the gas production rate, appears to be about10,000 times higher than what is measured on shale coresin the laboratory has been basically correct. However, thesefractures are completely closed and do not convey any gas,and their role is indirect.

vii) (i) No porosity⇒ no branching. (ii) No seepage forces⇒ nobranching. (iii) No weak layers⇒ no branching. (iv) ConstantBiot coefficient ⇒ no branching. (Note that, consequently,hydraulic crack branching in granite is impossible.)

ACKNOWLEDGMENTS. Z.P.B. thanks Emmanuel Detournay for valuable dis-cussions and a stimulating challenge that provoked part of this study. Thework at Northwestern was funded by Los Alamos National Laboratory(Grants DOE LDRD 20170103DR and OBES DE-AC52-06NA25396) to North-western University. Z.P.B. is McCormick Institute Professor and W. P. MurphyProfessor of Civil and Mechanical Engineering and Materials Science atNorthwestern University.

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